In order to find an expansion which can be used for small \(S_{i}\) , we
apply the product theorem for Bessel
functions [1]}
to (),
\(\frac{H_{1}^{(1)}(\alpha \sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}&=2\sum _{q=0}^{\infty }(-1)^{q}(2q+1)H_{2q+1}^{(1)}(\alpha )\frac{J_{2q+1}(\alpha u)}{\alpha u}.\) 
